We employ the theory of spatial point processes to revisit and reinterpret a particular class of time-variant stochastic radio channel models. Common for all models in this class is that individual multipath components are emerging and vanishing in a temporal birth-death like manner, with the underlying stochastic birth-death mechanism governed by two facilitating assumptions. Well-known analytical properties of this class of channel models are reestablished by simple arguments and several new results are derived. The primary tool used to obtain these results is Campbell's Theorem which enables novel assessment of the autocorrelation functions of random processes used in the general channel model description. Under simplifying assumptions the channel transfer function is shown to be wide-sense stationary in both time and frequency (despite the birth-death behavior of the overall channel). The proof of this result is a consequence of the point process perspective, in particular by circumventing enumeration issues arising from the use of integer-indexed path components in traditional channel modeling approaches. The practical importance of being able to analytically characterize the birth-death channel models is clearly evidenced, e.g., by the fact that key parameters enter explicitly in measurable quantities such as the power-delay profile.